金融工程中的蒙特卡罗方法 影印版PDF|Epub|txt|kindle电子书版本网盘下载

金融工程中的蒙特卡罗方法 影印版PDF格式电子书版下载

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1 Foundations1

1.1 Principles of Monte Carlo1

1.1.1 Introduction1

1.1.2 First Examples3

1.1.3 Efficiency of Simulation Estimators9

1.2 Principles of Derivatives Pricing19

1.2.1 Pricing and Replication21

1.2.2 Arbitrage and Risk-Neutral Pricing25

1.2.3 Change of Numeraire32

1.2.4 The Market Price of Risk36

2 Generating Random Numbers and Random Variables39

2.1 Random Number Generation39

2.1.1 General Considerations39

2.1.2 Linear Congruential Generators43

2.1.3 Implementation of Linear Congruential Generators44

2.1.4 Lattice Structure47

2.1.5 Combined Generators and Other Methods49

2.2 General Sampling Methods53

2.2.1 Inverse Transform Method54

2.2.2 Acceptance-Rejection Method58

2.3 Normal Random Variables and Vectors63

2.3.1 Basic Properties63

2.3.2 Generating Univariate Normals65

2.3.3 Generating Multivariate Normals71

3 Generating Sample Paths79

3.1 Brownian Motion79

3.1.1 One Dimension79

3.1.2 Multiple Dimensions90

3.2 Geometric Brownian Motion93

3.2.1 Basic Properties93

3.2.2 Path-Dependent Options96

3.2.3 Multiple Dimensions104

3.3 Gaussian Short Rate Models108

3.3.1 Basic Models and Simulation108

3.3.2 Bond Prices111

3.3.3 Multifactor Models118

3.4 Square-Root Diffusions120

3.4.1 Transition Density121

3.4.2 Sampling Gamma and Poisson125

3.4.3 Bond Prices128

3.4.4 Extensions131

3.5 Processes with Jumps134

3.5.1 A Jump-Diffusion Model134

3.5.2 Pure-Jump Processes142

3.6 Forward Rate Models: Continuous Rates149

3.6.1 The HJM Framework150

3.6.2 The Discrete Drift155

3.6.3 Implementation160

3.7 Forward Rate Models: Simple Rates165

3.7.1 LIBOR Market Model Dynamics166

3.7.2 Pricing Derivatives172

3.7.3 Simulation174

3.7.4 Volatility Structure and Calibration180

4 Variance Reduction Techniques185

4.1 Control Variates185

4.1.1 Method and Examples185

4.1.2 Multiple Controls196

4.1.3 Small-Sample Issues200

4.1.4 Nonlinear Controls202

4.2 Antithetic Variates205

4.3 Stratified Sampling209

4.3.1 Method and Examples209

4.3.2 Applications220

4.3.3 Poststratification232

4.4 Latin Hypercube Sampling236

4.5 Matching Underlying Assets243

4.5.1 Moment Matching Through Path Adjustments244

4.5.2 Weighted Monte Carlo251

4.6 Importance Sampling255

4.6.1 Principles and First Examples255

4.6.2 Path-Dependent Options267

4.7 Concluding Remarks276

5 Quasi-Monte Carlo281

5.1 General Principles281

5.1.1 Discrepancy283

5.1.2 Van der Corput Sequences285

5.1.3 The Koksma-Hlawka Bound287

5.1.4 Nets and Sequences290

5.2 Low-Discrepancy Sequences293

5.2.1 Halton and Hammersley293

5.2.2 Faure297

5.2.3 Sobol303

5.2.4 Further Constructions314

5.3 Lattice Rules316

5.4 Randomized QMC320

5.5 The Finance Setting323

5.5.1 Numerical Examples323

5.5.2 Strategic Implementation331

5.6 Concluding Remarks335

6 Discretization Methods339

6.1 Introduction339

6.1.1 The Euler Scheme and a First Refinement339

6.1.2 Convergence Order344

6.2 Second-Order Methods348

6.2.1 The Scalar Case348

6.2.2 The Vector Case351

6.2.3 Incorporating Path-Dependence357

6.2.4 Extrapolation360

6.3 Extensions362

6.3.1 General Expansions362

6.3.2 Jump-Diffusion Processes363

6.3.3 Convergence of Mean Square Error365

6.4 Extremes and Barrier Crossings: Brownian Interpolation366

6.5 Changing Variables371

6.6 Concluding Remarks375

7 Estimating Sensitivities377

7.1 Finite-Difference Approximations378

7.1.1 Bias and Variance378

7.1.2 Optimal Mean Square Error381

7.2 Pathwise Derivative Estimates386

7.2.1 Method and Examples386

7.2.2 Conditions for Unbiasedness393

7.2.3 Approximations and Related Methods396

7.3 The Likelihood Ratio Method401

7.3.1 Method and Examples401

7.3.2 Bias and Variance Properties407

7.3.3 Gamma411

7.3.4 Approximations and Related Methods413

7.4 Concluding Remarks418

8 Pricing American Options421

8.1 Problem Formulation421

8.2 Parametric Approximations426

8.3 Random Tree Methods430

8.3.1 High Estimator432

8.3.2 Low Estimator434

8.3.3 Implementation437

8.4 State-Space Partitioning441

8.5 Stochastic Mesh Methods443

8.5.1 General Framework443

8.5.2 Likelihood Ratio Weights450

8.6 Regression-Based Methods and Weights459

8.6.1 Approximate Continuation Values459

8.6.2 Regression and Mesh Weights465

8.7 Duality470

8.8 Concluding Remarks478

9 Applications in Risk Management481

9.1 Loss Probabilities and Value-at-Risk481

9.1.1 Background481

9.1.2 Calculating VAR484

9.2 Variance Reduction Using the Delta-Gamma Approximation492

9.2.1 Control Variate493

9.2.2 Importance Sampling495

9.2.3 Stratified Sampling500

9.3 A Heavy-Tailed Setting506

9.3.1 Modeling Heavy Tails506

9.3.2 Delta-Gamma Approximation512

9.3.3 Variance Reduction514

9.4 Credit Risk520

9.4.1 Default Times and Valuation520

9.4.2 Dependent Defaults525

9.4.3 Portfolio Credit Risk529

9.5 Concluding Remarks535

A Appendix: Convergence and Confidence Intervals539

A.1 Convergence Concepts539

A.2 Central Limit Theorem and Confidence Intervals541

B Appendix: Results from Stochastic Calculus545

B.1 Ito's Formula545

B.2 Stochastic Differential Equations548

B.3 Martingales550

B.4 Change of Measure553

C Appendix: The Term Structure of Interest Rates559

C.1 Term Structure Terminology559

C.2 Interest Rate Derivatives564

References569

Index587